We study the problem of anyons with statistics \ensuremath{\nu} in a strong magnetic field by means of a similarity transformation. The ground state of this problem exhibits an ``integer'' quantum Hall effect at filling factor 1/\ensuremath{\nu} with quasiparticle excitations of charge q/\ensuremath{\nu} and statistics -1/\ensuremath{\nu}. Certain electron fractional quantum Hall states can be considered as a realization of this, for example, the 2/5, 3/7,. . . hierarchy of the 1/3 state. This may explain the observed quasiparticle-quasihole asymmetry in the fractional quantum hall hierarchy.
The ground state of an ideal gas of anyons with statistical angle
θ∈(0,π) is a long-standing open problem, with renewed interest from the recent observations of fractional Chern insulators. In this work we construct variational wavefunctions that build in the exact anyon statistics and combine theory-guided composite-boson, composite-fermion and parton wavefunctions with neural-network correlations on the sphere. For spinless and SU(2) semions we confirm the paired anyon superfluid, well below the best crystal states. At
θ=π/3 liquids and crystals are closer in energy for the spinless case. For
θ=π/3 anyons with SU(3) flavor, we find two candidates. One is a skyrmion Hall crystal, in which the anyons form an integer quantum Hall state in the effective magnetic field from the spin texture. The other is a chiral color superconductor of six-anyon clusters, the color-flavor-locked pairing descendant of a holon metal with SU(3) color gauge symmetry and SU(3) flavor symmetry; its wavefunction is a hyperdeterminant rather than a product of parton determinants. Within our variational ansatz the color superconductor has better energy than both crystals and the color-flavor locked phases with U(1)
2 and Z
3 gauge groups. For twisted MoTe
2 this suggests that the reentrant integer quantum anomalous Hall phase is the skyrmion Hall crystal of the doped anyons, and that the observed superconductivity may be the color superconductor, which can undergo a transition to an
f+if BCS superconductor with a jump of the chiral central charge by 3.
We present an exact scheme of bosonization for anyons (including fermions) in the two-dimensional manifold of the quantum Hall fluid. This gives every fractional quantum Hall phase of the electrons one or more dual bosonic descriptions. For interacting electrons, the statistical transmutation from anyons to bosons allows us to explicitly derive the microscopic statistical interaction between the anyons, in the form of the effective two-body and few-body interactions. This also leads to a number of unexpected topological phases of the single component bosonic fractional quantum Hall effect that may be experimentally accessible. Numerical analysis of the energy spectrum and ground state entanglement properties are carried out for simple examples.